Phase diagram. The topological magnetic phase diagram for a set of model parameters is shown in Fig. REF . The FM and non-FM regions are highlighted in the \((D_2/J_1,D_1/J_1)\)  plane. The zero temperature phase diagram was constructed via energy comparison between non-ferromagnetic versus ferromagnetic states (which have the lowest energy). The topological phases, characterized by Chern numbers [1]}, are shown in Fig. REF . The bulk-boundary correspondence can be seen in Fig. REF . According to Fig. REF , the Chern number combination is \(\lbrace 1,-1,-1,1\rbrace \) . Since the winding numbers are \(w_1=-w_3=1\) , \(w_2=0\) , there is only one loop of edge state in the first and the third gap with opposite orientation. Hence, there is no edge state in the second gap. The group velocity of the edge states show a chirally moving magnon current localized at the edges, whose direction is consistent with the total winding number in that gap. A thorough phase diagram calculation is reported in the Supplementary Information.
